dan's prealgebra lesson

A friend of mine once said, "I had algebra, it's that 'a + b = c' stuff." When I asked, the friend couldn't explain what a, b, and c were, what they might have meant, or why you would even "add letters" in the first place. Patty Leitner and I decided to write a book about this very thing:

Prealgebra, Mathematics for a Variable World, by Daniel Bach and Patricia Leitner, Third edition, McGraw-Hill, 2006.

Variables

The main idea is that in algebra, a variable represents a number (or name, etc.) whose value might vary; hence the name!

Example: My sister Emily is 4 years older than me, so:

When I was 10, she was 10 + 4 = 14 .

When I was 17, she was 17 + 4 = 21 .

When I was (Dan's age), she was (Dan's age) + 4 .

We can say (Emily's age) = (Dan's age) + 4 ,

or simply E = D+ 4 , where E = Emily's age, and D = Dan's age .

The quantities "Dan's age", "Emily's age", "D", and "E" are variables because they can represent many different numbers.

Laws of Arithmetic

You know 3 + 5 = 8. Does it matter what order you add?

No; 5 + 3 = 8 too. So 3 + 5 = 5 + 3.

Also (20) + (–3) = (–3) + (20); both are 17.

So we can pretty much say that

a + b = b + a , for any numbers a and b.

This is the commutative law of addition. (when you commute, you go back and forth.)